What if a right-handed person is in the hospital and has bandages on his left arm- can he put his Shel Yad on his right arm?
He may not put his tefillin on his right arm, because that is not his weaker arm. Putting it on the arm is like putting it on his foot, it is just not the correct place for it. If he can manage to at least put the tefillin on the upper part of his arm, where the boxes go, then he has fulfilled the mitzva. If that part is covered with bandages, he should still put the tefillin over the bandages, if that is possible, but without saying the bracha. Then when he puts on the shel rosh he will make both brachos (for Ashkenazim, but Sefardim will only say al mitzvas tefillin).
Chelkas Yackov 2-43, Biur Halacha 27-1 D:H B’lo Bracha, PIskei Teshuvos 27-11.